Semilocal convergence for Newton’s method on a Banach space with a convergence structure and twice Fréchet differentiable operators.

*(English)*Zbl 1033.65035The paper deals with the problem of approximating a locally unique solution of a equation \(F(x)=O\) in a Banach space with a convergence structure. The operators are assumed to be twice Fréchet differentiable. Semilocal convergence is proved for the Newton-Kantorovich method. If the Banach space is equipped with a real norm some known results are derived.

Moreover, the author compares his results favorably with earlier results involving only once Fréchet differentiable operators because the sufficient convergence condition given in this paper can be weaker than the usual one. A polynomial equation is studied as an example, where the Kantorovich hypothesis (assumptions of the convergence theorem) fails, the presented theorem guarantees the local convergence.

Moreover, the author compares his results favorably with earlier results involving only once Fréchet differentiable operators because the sufficient convergence condition given in this paper can be weaker than the usual one. A polynomial equation is studied as an example, where the Kantorovich hypothesis (assumptions of the convergence theorem) fails, the presented theorem guarantees the local convergence.

Reviewer: Werner H. Schmidt (Greifswald)

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |

47J25 | Iterative procedures involving nonlinear operators |